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The surface area of a shape is the total area of all of the shape's faces. It can be considered the total extent of all of the shape's 2-subfacets in 2-dimensional space. The surface area of a polygon is typically just called its area.

A surface area has dimensions of [length]2.

Surface Area Formulae

2-Dimensional

Shape Area Formula Variables
Square ab a, b = edge lengths
Disk \frac{\pi}{4} d^2 d = diameter of disc
Regular P-gon \frac{p}{4} \cot{\left( \frac{\pi}{p} \right)} a^2 p = number of edges, a = edge length
Regular P/Q-gon \frac{p}{4} \cot{\left( \frac{\pi q}{p} \right)} a^2 p = number of edges, q = winding number, a = edge length
Triangle \frac{\sqrt{3}}{4} a^2 a = edge length
Pentagon \frac{1}{4} \sqrt{5 \left( 5 + 2 \sqrt{5} \right)} a^2 a = edge length
Hexagon \frac{3 \sqrt{3}}{2} a^2 a = edge length
Octagon 2\left(1 + \sqrt{2}\right) a^2 a = edge length
Decagon \frac{5}{2} \sqrt{ 5 + 2 \sqrt{5} } a^2 a = edge length
Dodecagon 3\left(2 + \sqrt{3}\right) a^2 a = edge length
Pentadecagon \frac{15}{4} \sqrt{7 + 2\sqrt{5} + 2 \sqrt{3 \left( 5 + 2 \sqrt{5} \right) } }  a^2 a = edge length
Hexadecagon 4 \left( 1 + \sqrt{2} + \sqrt{ 2 \left( 2 + \sqrt{2} \right) }  \right) a^2 a = edge length
Icosagon 5 \left( 1 + \sqrt{5} + \sqrt{5 + 2\sqrt{5}} \right) a^2 a = edge length
Icositetragon 6 \left( 2 + \sqrt{6} + \sqrt{5 + 2\sqrt{6}} \right) a^2 a = edge length
Triacontagon \frac{15}{2} \sqrt{23 + 10\sqrt{5} + 2 \sqrt{3 \left( 85 + 38\sqrt{5} \right)} } a^2 a = edge length
Triacontadigon 8 \sqrt{\frac{ 2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 - \sqrt{2 + \sqrt{2 + \sqrt{2}}} }} a^2 a = edge length
Tube \pi ad a = edge length, d = diameter of circle
Torus \frac{1}{16} \pi^2 d_1 d_2 d1, d2 = diameters of circles
Sphere \pi d^2 d = diameter of sphere

3-Dimensional

Shape Area Formula Variables
Cube 2\left(ab + ac + bc\right) a, b, c = edge lengths
Cylinder \pi d \left( \frac{1}{2} d + a \right) a = edge length, d = diameter of disk
Ball \pi d^2 d = diameter of ball
Tetrahedron \sqrt{3} a^2 a = edge length
Octahedron 2\sqrt{3} a^2 a = edge length
Dodecahedron 3 \sqrt{ 5 \left( 5 + 2 \sqrt{5} \right) } a^2 a = edge length
Icosahedron 5\sqrt{3} a^2 a = edge length
Regular P,Q-hedron \frac{ pq \cot{\left( \frac{\pi}{p} \right)}  } { 4-\left( p-2 \right) \left( q - 2 \right) } a^2 a = edge length
Solid torus \frac{1}{16} \pi^2 d_1 d_2 d1, d2 = diameters of circle & disk

Surface Areas

The following are surface areas of some physical objects

Surface Area Object
2.6×10-70 m2 Planck area
9.0×1012 m2 Australia
1.0×1013 m2 Europe
1.4×1013 m2 Antarctica
1.8×1013 m2 South America
2.5×1013 m2 North Ameria
3.0×1013 m2 Africa
4.2×1013 m2 America
4.4×1013 m2 Asia
5.4×1013 m2 Eurasia
8.4×1013 m2 Afro-Eurasia
1.5×1014 m2 Earth's land area
5.1×1014 m2 Earth

Type 1.0 Civilization

6.1×1018 m2 the Sun
2.0×1032 m2 the Oort Cloud
7.0×1041 m2 Disk of the Milky Way

Type 3.0 Civilization

2.4×1054 m2 Observable Universe

Type 4.0 Civilization

See Also

Space Antiderivatives
Vertex count · Edge length · Surface area · Surcell volume · Surteron bulk · Surpeton pentavolume · Surecton hexavolume · Surzetton heptavolume · Suryotton octavolume

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